Evolution Equations and Control Theory (EECT)

Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions

Pages: 631 - 667, Volume 2, Issue 4, December 2013      doi:10.3934/eect.2013.2.631

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Nicolas Fourrier - Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States (email)
Irena Lasiecka - Department of Mathematics, University of Memphis, Memphis, TN 38152-3370, IBS, Polish Academy of Sciences, Warsaw, Poland (email)

Abstract: We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both.
    This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well-posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed.
    The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.

Keywords:  Wave equation with dynamic boundary conditions, strong damping, Laplace'a Beltrami operator, semigroup generation, analyticity of semigroups, Gevrey's regularity, strong stability, uniform stability, spectral analysis.
Mathematics Subject Classification:  Primary: 35Lxx, 35Kxx, 35Pxx; Secondary: 65kxx, 76Nxx.

Received: April 2013;      Revised: September 2013;      Available Online: November 2013.